A man on the deck of a ship is 16 m above the water level. He observes that the angle of elevation of the top of a cliff is 45° and the angle of depression of the base is 30°. Calculate the distance of the cliff from the ship and the height of the cliff.

Let A be the man on the deck of the ship B and CE is the cliff.

AB = 16 m and angle of elevation of the top of cliff is 45° and angle of depression of base of cliff is 30°.

Let CE = h, AD = x, then
CD = h - 16, AD = BE = x.

Now in right angled triangle △CAD,

$\text{tan 45°} = \dfrac{CD}{AD} \\[1em] 1 = \dfrac{h - 16}{x} \\[1em] x = h - 16 \text{ ……(i)}$

Again in right angled triangle △ADE,

$\text{tan 30°} = \dfrac{DE}{AD} = \dfrac{16}{x} \\[1em] \dfrac{1}{\sqrt{3}} = \dfrac{16}{x} \\[1em] x = 16\sqrt{3} = 27.71 \text{ m}$

From (i) and (ii),

$\Rightarrow h - 16 = 27.71 \\[1em] \Rightarrow h = 27.71 + 16 \\[1em] \Rightarrow h = 43.71 \text{ m}$

Hence, the distance of cliff from the ship is 27.71 m and height of cliff is 43.71 m.